Steady State Diffusion
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In steady state, there is no change in concentration so ∂c/∂t = 0
and Fick's second law says 0 = ∇⋅J. For an inert
flat layer, thickness L, this means a linear profile and a flux directly
proportional to the concentration difference Δc across the layer:
which is used in experiments to measure diffusion parameters.
In biology, there can be chemical conversion, and steady state implies that this
process has to be matched by supply. For tissue with an oxygen (O2)
consumption M, the changes due to consumption M and
diffusion – Fick's second law – must cancel out:
0 = − M − ∇⋅J
and combination with Fick's first law leads to:
0 = − M + ℘∇2P
where now the form with partial pressure P is used because it is the gas
oxygen that has to supply the tissue. Again, for a flat layer:
But more interesting is the famous Krogh-Erlang equation
for O2 diffusing out of a central capillary into a surrounding circular
(2-dimensional) region:
| P = Pc + |
M |
{
r2 − rc2 − R2
ln( |
r2 |
)} |
| 4℘ |
rc2 |
where Pc is capillary rim O2 pressure, r distance from the center,
where the capillary is, with radius rc, and R radius of the circular region.
For more about that choose the ‘oxygen transport in muscle tissue’
item in the “Career” page.
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